GLOBAL DYNAMICS AND UNFOLDING OF PLANAR

PIECEWISE SMOOTH QUADRATIC QUASI–HOMOGENEOUS

DIFFERENTIAL SYSTEMS

YILEI TANG1,2

Abstract. In this paper we research global dynamics and bifurcations ofplanar piecewise smooth quadratic quasi–homogeneous but non–homogeneouspolynomial differential systems. We present sufficient and necessary conditionsfor the existence of a center in piecewise smooth quadratic quasi–homogeneoussystems. Moreover, the center is global and non–isochronous if it exists,which cannot appear in smooth quadratic quasi–homogeneous systems. Thenthe global structures of piecewise smooth quadratic quasi–homogeneous butnon–homogeneous systems are studied. Finally we investigate limit cycle bi-furcations of the piecewise smooth quadratic quasi–homogeneous center andgive the maximal number of limit cycles bifurcating from the periodic orbitsof the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.

1. Introduction

Since Andronov et al [4] researched the properties of solutions of piecewiselinear differential systems, there are lots of works in mechanics, electrical engi-neering and the theory of automatic control which are described by non-smoothsystems; see for the works of Filippov [12], di Bernardo et al [7], Makarenkov andLamb [30] and the references therein.

For the planar piecewise smooth linear differential systems separated by astraight line, [9, 20, 28] studied the systems having two or three limit cycles re-spectively. More investigations of limit cycle bifurcations from linear piecewisedifferential systems can be seen in [13, 16]. The discussion of limit cycle bifur-cations in nonlinear piecewise differential equations has also been researched inmany works; see for instance [8, 10, 25, 33, 34].

However, there are seldom works giving completely global dynamics of piece-wise smooth nonlinear differential systems. Even for smooth polynomial differ-ential systems there are only few classes whose global structures were completelycharacterized, as shown in [11, 31].

A real planar polynomial differential system

x = P (x, y), y = Q(x, y),(1.1)

2010 Mathematics Subject Classification. Primary: 37G05, Secondary: 37G10, 34C23, 34C20.Key words and phrases. Quasi–homogeneous polynomial systems; global phase portrait; bi-

furcation of piecewise system; Melnikov function.

1

2 Y. TANG

is called a quasi–homogeneous polynomial differential system if there exist con-stants s1, s2, d ∈ Z+ such that for an arbitrary α ∈ R+ it holds that

P (αs1x, αs2y) = αs1+d−1P (x, y), Q(αs1x, αs2y) = αs2+d−1Q(x, y),

where PQ ≡ 0, P (x, y), Q(x, y) ∈ R[x, y], Z+ is the set of positive integers andR+ is the set of positive real numbers. We denominate w = (s1, s2, d) the weightvector of system (1.1) or of its associated vector field. When s1 = s2 = 1, system(1.1) is a homogeneous one of degree d. Clearly, quasi–homogeneous system (1.1)

has a unique minimal weight vector (MWF for short) w = (s1, s2, d) satisfying

s1 ≤ s1, s2 ≤ s2 and d ≤ d for any other weight vector (s1, s2, d) of system (1.1).We say that system (1.1) has degree n if n = max{degP, degQ}. In what followswe assume without loss of generality that P and Q in system (1.1) have not anon–constant common factor.

Smooth Quasi–homogeneous polynomial differential systems have been inten-sively studied by a great deal of authors from different views. We refer readersto see for example the integrability [2, 17, 19, 21, 29], the centers and limit cy-cles [1, 15, 18, 24], the algorithm to compute quasi–homogeneous systems witha given degree [14], the characterization of centers or topological phase portraitsfor quasi–homogeneous equations of degrees 3-5 respectively [5, 26, 32] and thereferences therein.

A real planar piecewise smooth polynomial differential system

x = P+(x, y), y = Q+(x, y), y ≥ 0,x = P−(x, y), y = Q−(x, y), y < 0

(1.2)

is called a piecewise smooth quasi–homogeneous polynomial differential systemwith two zones separated by the x-axis if both (P+(x, y), Q+(x, y)) and (P−(x, y),Q−(x, y)) are quasi–homogeneous polynomial vector fields.

In this paper we research the global dynamics and bifurcations of all piecewisesmooth quadratic quasi–homogeneous but non–homogeneous differential systems.First the existence of a global and non-isochronous center at the origin of piecewisesmooth quadratic quasi–homogeneous but non–homogeneous systems is proved.Notice that the origin of smooth quadratic quasi–homogeneous systems cannotbe a center. Then we characterize the global phase portraits of piecewise smoothquadratic quasi–homogeneous but non–homogeneous polynomial vector fields. Atlast we perturb the piecewise smooth quadratic quasi–homogeneous system at thecenter by generic piecewise polynomials of degree n, and determine the maximalnumber of limit cycles bifurcating from the periodic orbits of the center by usingthe first order Melnikov function.

This article is organized as follows. In section 2 we prove that only one classof piecewise smooth quadratic quasi–homogeneous but non–homogeneous differ-ential systems has a center at the origin, and it is global and non-isochronous ifit exists. Section 3 will concentrate on global structures and phase portraits ofpiecewise smooth quadratic quasi–homogeneous but non–homogeneous differen-tial systems. The unfoldings and bifurcations of these systems are investigated

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 3

for some critical values of parameters. The last section is devoted to the lim-it cycle bifurcations from the periodic orbits of the piecewise smooth quadraticquasi–homogeneous center.

2. Center of piecewise smooth quadratic quasi–homogeneoussystems

Due to Proposition 17 of Garcıa, Llibre and Perez del Rıo [14], a smooth quasi-homogeneous but non-homogeneous quadratic system has one of the followingthree forms:

(i) : x = a1y2, y = b1x with MWV (3, 2, 2) and a1b1 = 0,

(ii) : x = a2xy, y = b21x+ b22y2 with MWV (2, 1, 2) and a2b21b22 = 0,

(iii) : x = a31x+ a32y2, y = b3y with MWV (2, 1, 1) and a31a32b3 = 0.

Thus after taking appropriate linear changes of variable x together with a timescaling, we have three totally reduced piecewise smooth quasi-homogeneous butnon-homogeneous quadratic systems.

Lemma 1. Every planar piecewise smooth quasi-homogeneous but non-homogeneousquadratic system is one of the following three systems:

(I) : x = a1y2, y = b1x if y ≥ 0,

x = a1y2, y = x if y < 0;

(II) : x = a2xy, y = b21x+ b22y2 if y ≥ 0,

x = a2xy, y = x+ y2 if y < 0;(III) : x = a31x+ a32y

2, y = b3y if y ≥ 0,x = a31x+ y2, y = y if y < 0,

where all parameters cannot be zero.

Proof. From the transformation (x, y, dt) → (x, y, b1dt), (x, y, dt) → (b21x/b22,

y, b22dt) and (x, y, dt) → (b3x/a32, y, b3dt), planar piecewise smooth quadraticquasi-homogeneous but non-homogeneous systems

(a) : x = a1y2, y = b1x if y ≥ 0,

x = a1y2, y = b1x if y < 0;

(b) : x = a2xy, y = b21x+ b22y2 if y ≥ 0,

x = a2xy, y = b21x+ b22y2 if y < 0;

(c) : x = a31x+ a32y2, y = b3y if y ≥ 0,

x = a31x+ a32y2, y = b3y if y < 0,

are changed into systems (I), (II) and (III) respectively, where we still write

a1/b1, b1/b1, a1/b1, a2/b22, b21/b21, b22/b22, a2/b22, a31/b3, a32/a32, b3/b3 and

a31/b3 as a1, b1, a1, a2, b21, b22, a2, a31, a32, b3 and a31 for simpler notations. �

4 Y. TANG

In the following, we briefly present the Filippov convex method [7, 12, 22, 23]to study the dynamics of generic piecewise smooth quasi-homogeneous system(1.2) close to the discontinuous line. This discontinuous line

L := {(x, y) ∈ R2| F (x, y) := y = 0}separates the plane into two open nonoverlapping regions

Y + = {(x, y) ∈ R2| 0 < y} and Y − = {(x, y) ∈ R2| 0 > y}.Suppose that

σ(x, y) =⟨(Fx, Fy), (P

+, Q+)⟩ ⟨

(Fx, Fy), (P−, Q−)

⟩,

where < ·, · > denotes the standard scalar product. The crossing set can bedefined by

Lc := {(x, y) ∈ L| σ(x, y) > 0},(2.1)

indicating that at each point of Lc the orbit of system (1.2) crosses L, i.e., theorbit reaching (x, y) from Y + (or Y −) concatenates with the orbit entering Y −

(or Y +) from (x, y). The sliding set Ls is the complement of Lc in L, which isdefined as

Ls := {(x, y) ∈ L| σ(x, y) ≤ 0}.(2.2)

Moreover, in Ls solving the equation⟨(Fx, Fy), (P

− − P+, Q− −Q+)⟩= 0(2.3)

we can obtain the singular sliding points from the set of solutions.Regarding to the piecewise smooth system (I), we can analyze that the crossing

set and the sliding set in L are

LIc = {(x, y) ∈ L| b1x2 > 0} =

{ {(x, y) ∈ L| x = 0} if b1 > 0,∅ if b1 < 0

(2.4)

and

LIs = {(x, y) ∈ L| b1x2 ≤ 0} =

{ {(x, y) ∈ L| x = 0} if b1 > 0,L if b1 < 0

(2.5)

respectively by definitions (2.1) and (2.2). Then, we find the only solution of(2.3) for system (I) in LI

s is the origin, which is a singular sliding point and atthe same time a boundary equilibrium because of the vanish of vector fields atthe origin.

By an analogous computation of system (I), we have the crossing sets and thesliding sets in L of the forms

LIIc = {(x, y) ∈ L| b21x2 > 0} =

{ {(x, y) ∈ L| x = 0} if b21 > 0,∅ if b21 < 0,

(2.6)

LIIs = {(x, y) ∈ L| b21x2 ≤ 0} =

{ {(x, y) ∈ L| x = 0} if b21 > 0,L if b21 < 0

(2.7)

and

(2.8) LIIIc = ∅, LIII

s = L

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 5

for piecewise smooth systems (II) and (III), respectively. We find the originof system (II) in LII

s is a unique singular sliding point, which is a boundaryequilibrium. Moreover, the discontinuous line L is full of non-isolated singularsliding points for system (III), since equation (2.3) always holds on the slidingset LIII

s .

Notice that all smooth quadratic quasi–homogeneous systems (i) − (iii) haveno centers, since there exists an invariant line or an invariant curve passingthrough the origin of such systems. However, for piecewise smooth quadraticquasi–homogeneous systems we will find the existence of a center at the originunder some parameter conditions.

An equilibrium of the piecewise smooth system (1.2) is called a center if al-l solutions sufficiently closed to it are periodic. If all periodic solutions insidethe period annulus of the center have the same period it is said that the cen-ter is isochronous. A center is called a global center when the periodic orbitssurrounding the center fill the whole plain except the center itself.

Theorem 2. Piecewise smooth quadratic quasi–homogeneous systems (II) and(III) have no centers on the phase space. Piecewise smooth quadratic quasi–homogeneous system (I) has a center at the origin if and only if a1 < 0, b1 > 0and a1 > 0, which is global but not isochronous.

Proof. Notice that no equilibria of piecewise smooth systems (I)-(III) exist in theregions Y ±. From above mentioned analysis of sliding sets and singular slidingpoints, on the discontinuous line L systems (I) and (II) have a unique singularsliding point at the origin, and L is full of non-isolated singular sliding points forsystem (III). Thus, system (III) has no centers on the plane. It is easy to seethat system (II) has an invariant line x = 0 passing through its origin, yieldingthat the origin cannot be a center. We only need to check whether the origin ofsystem (I) can be a center.

The corresponding smooth quadratic quasi–homogeneous system (i) of piece-wise smooth system (I) has a double vanished eigenvalue and by [11, Theorem3.5] the equilibrium at the origin is a cusp. Moreover, when a1b1 > 0 the piece-

wise smooth system (I) has an invariant curve a13 y

3 − b12 x

2 = 0 passing throughthe origin O1 : (0, 0) in the half plane Y +, and when a1 < 0 system (I) has aninvariant curve a1

3 y3 − 1

2x2 = 0 passing through the origin O1 in the half plane

Y −. Therefore, only when a crossing deleted neighborhood (−δ0, δ0) \ {0} existsfor small δ0 > 0 under the condition a1b1 < 0, a1 > 0, the origin O1 of system(I) is possibly a center. Thus we get the parameter condition a1 < 0, b1 > 0 anda1 > 0 from the expression of the crossing set LI

c of piecewise smooth system (I).When a1 < 0, b1 > 0 and a1 > 0, the crossing set LI

c is the x-axis except theorigin and the orbits surrounding the origin are spirals rotating anti-clockwise.Let p+(r, θ) (resp. p−(r, θ)) be the solution of piecewise smooth system (I) inpolar coordinates (x, y) = (ρ cos θ, ρ sin θ) for 0 ≤ θ < π (resp. −π ≤ θ < 0),satisfying that the initial condition p+(r, 0) = r (resp. p−(r,−π) = r) holds,which is well defined in the region R2 \LI

s. Then, we define the positive Poincare

6 Y. TANG

half-return map as P+(r) := limθ→π p+(r, θ) and the negative Poincare half-

return map as P−(r) := limθ→0 p−(r, θ), as shown in Figure 1. The Poincare

return map associated to piecewise smooth system (I) is given by the compositionof these two maps

(2.9) PI(r) := P−(P+(r)).

In order to obtain the existence of a center and further a global center at theorigin, via the definition (2.9) we need to prove PI(r)− r ≡ 0 for arbitrary r > 0.

O

r

Figure 1. Existence of closed orbits for system (I).

Piecewise smooth system (I) has a polynomial first integral H+1 (x, y) = a1

3 y3−

b12 x

2 if y ≥ 0, and a first integral H−1 (x, y) = a1

3 y3 − 1

2x2 if y < 0. Then we have

H+1 (r, 0) = H+

1 (P+(r), 0), yielding that P+(r) = r. Furthermore, by

H−1 (P+(r), 0) = H−

1 (PI(r), 0)

we get PI(r) = r, implying that the solution curve of piecewise smooth system(I) through (r, 0) is a closed orbit for arbitrary r > 0. Notice that the origin O1

is the unique singularity of piecewise smooth system (I) when a1 < 0, b1 > 0 anda1 > 0. Therefore, the origin O1 of piecewise smooth system (I) is a center if andonly if a1 < 0, b1 > 0 and a1 > 0 and furthermore it is a global center.

Next, in the case a1 < 0, b1 > 0 and a1 > 0 we prove that the center O1 atthe origin of piecewise smooth system (I) is not isochronous. Assuming that Γr0

is the closed trajectory through (r0, 0) inside the periodic annulus of the centerO1, we can define the positive half-period function as T+(r0) :=

∫Γ+r0

dt and the

negative half-period function as T−(r0) :=∫Γ−r0

dt, where r0 > 0,

Γ+r0 = {(ρ, θ) ∈ R2 | ρ = p+(r0, θ)} = {(x, y) ∈ R2 | y =

( 3b12a1

(x2 − r20)) 1

3 }

and

Γ−r0 = {(ρ, θ) ∈ R2 | ρ = p−(r0, θ)} = {(x, y) ∈ R2 | y =

( 3

2a1(x2 − r20)

) 13 }.

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 7

Thus the complete period function associated to piecewise smooth system (I) isgiven by the sum of these two functions

TI(r0) =

∮Γr0

dt = T+(r0) + T−(r0) =

∫Γ+r0

dx

a1y2+

∫Γ−r0

dx

a1y2

=

∫ −r0

r0

dx

a1

(3b12a1

(x2 − r20)) 2

3

+

∫ r0

−r0

dx

a1

(3

2a1(x2 − r20)

) 23

= β0r− 1

30 ,

where β0 =( 23)53 π

32√3

Γ( 23)Γ( 5

6)(−a

− 13

1 b− 2

31 + a

− 13

1 ) > 0 and the Gamma function Γ(z) =∫∞0 e−ssz−1 ds. Clearly the period TI(r0) of the periodic orbits inside the periodannulus of the center O1 is monotonic in r0 and then it cannot be isochronous.We complete the proof of the theorem. �

3. Global structures of piecewise smooth quadraticquasi–homogeneous systems

We will apply the ideal of Poincare compactification to study the global struc-tures of piecewise smooth quadratic quasi–homogeneous but non–homogeneoussystems. Although this theory is usually used in smooth systems, our strategyis to analyze the properties at infinity on half plane one by one, and via the dis-cussion of sliding sets and crossing sets we can summarize the global topologicalstructures of piecewise smooth quadratic quasi–homogeneous systems.

First we briefly remind the procedure of Poincare compactification [3, 11].Consider a planar vector field

X = P (x, y)∂

∂x+ Q(x, y)

∂

∂y,

where P (x, y) and Q(x, y) are polynomials of degree n. Set S2 = {y = (y1, y2, y3) ∈R3 : y21 + y22 + y23 = 1}, S1 be the equator of S2 and p(X ) be the Poincare com-pactification of X on S2. Note that S1 is invariant under the flow of p(X ).

We consider the six local charts Ui = {y ∈ S2 : yi > 0} and Vi = {y ∈S2 : yi < 0} where i = 1, 2, 3 for the calculation of the expression of p(X ).The diffeomorphisms Fi : Ui → R2 and Gi : Vi → R2 for i = 1, 2, 3 arethe inverses of the central projections from the planes tangent at the points(1, 0, 0), (−1, 0, 0), (0, 1, 0), (0,−1, 0), (0, 0, 1) and (0, 0,−1) respectively. We de-note by (u, z) the value of Fi(y) or Gi(y) for any i = 1, 2, 3. The expression forp(X ) in the local chart (U1, F1) is given by

u = zn[−uP

(1

z,u

z

)+ Q

(1

z,u

z

)], z = −zn+1P

(1

z,u

z

),

8 Y. TANG

for (U2, F2) is

u = zn[P

(u

z,1

z

)− uQ

(u

z,1

z

)], z = −zn+1Q

(u

z,1

z

),

and for (U3, F3) is

u = P (u, z), z = Q(u, z).

When we study the equilibria at infinitt on the charts U2 ∪ V2, we only need toverify if the origins of these charts are singular points.

Theorem 3. Piecewise smooth quadratic quasi–homogeneous systems (I), (II)and (III) have totally 8, 64 and 36 global phase portraits respectively.

Proof. For piecewise smooth system (I), the origin of the corresponding smoothsystem (i) is a cusp by the proof of Theorem 2. Moreover, in the half planey > 0 (resp. y < 0) the piecewise smooth system (I) has an invariant curvea13 y

3 − b12 x

2 = 0 (resp. a13 y

3 − 12x

2 = 0) passing through the origin O1 : (0, 0)when a1b1 > 0 (resp. a1 < 0).

Taking respectively the Poincare transformations x = 1/z, y = u/z in the localchart U1 and x = u/z, y = 1/z in the local chart U2, smooth system (i) aroundthe equator of the Poincare sphere can be written respectively in

u = b1z − a1u3, z = −a1u

2z(3.1)

and

u = a1 − b1u2z, z = −b1uz

2.

Then singularities at infinity of system (i) only exist on the x-axis, whose cor-responding equilibrium in the local chart U1 (the origin of (3.1)) is a nodeby [11, Theorem 3.5]. Besides, system (i) has the polynomial first integral

H+1 (x, y) = a1

3 y3− b1

2 x2. Therefore, it is not difficult to get global phase portraits

of smooth system (i).Notice that piecewise smooth system (I) is invariant under the change (x, y) →

(−x, y) after a time rescaling dτ → −dτ , so we only need to consider phaseportraits in the half plane x ≥ 0. From (2.4) and (2.5), we know that thecrossing set LI

c = {(x, y) ∈ R2| y = 0, x = 0} if b1 > 0 and the sliding setLIs = {(x, y) ∈ R2| y = 0} if b1 < 0. The origin O1 is the unique singular sliding

point of piecewise smooth system (I). Hence when b1 > 0, the global phaseportraits of piecewise smooth system (I) can be obtained by the global phaseportraits of system (i) in the half plane y > 0 and y < 0 respectively, wherefour cases b1 > 0, a1 > 0, a1 > 0; b1 > 0, a1 > 0, a1 < 0; b1 > 0, a1 < 0, a1 < 0and b1 > 0, a1 < 0, a1 > 0 are considered. Notice that the whole plane is theperiod annulus of the center at the origin of system (I) if b1 > 0, a1 < 0, a1 > 0.There exist infinitely many homoclinic loops connecting with the singularities atinfinity on the x-axis and an eight-shape heteroclinic loop connecting with thesingularities at infinity on the x-axis and the origin if b1 > 0, a1 > 0, a1 < 0. Incontrast, when b1 < 0 the whole x-axis is the sliding set and each point exceptthe origin on the x-axis is a ”colliding” point of orbits, i.e., the orbit connecting

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 9

the point from the half plane y > 0 is along the opposite direction with that fromthe half plane y < 0. Thus, neither closed orbits nor homoclinic loops could existif b1 < 0. For b1 < 0, we research the global phase portraits of piecewise smoothsystem (I) in four cases: a1 > 0, a1 > 0; a1 > 0, a1 < 0; a1 < 0, a1 < 0 anda1 < 0, a1 > 0. Note that in the case b1 < 0, a1 > 0, a1 > 0 it seems that theorigin is surrounded by closed orbits but it is not true, since the direction of upperhalf of each oval is clockwise but the direction of the lower half is anticlockwise.Thus the ovals existing in the case b1 < 0, a1 < 0, a1 > 0 are not homoclinicloops actually by the similar reason. Thus we obtain 8 global phase portraits forpiecewise smooth system (I) under above 8 parameter conditions.

We next investigate the piecewise smooth system (II) for its global structures.Using Theorem 3.5 of [11], the origin of smooth system (ii) is a saddle if a2b22 < 0,and its neighborhood consists of a hyperbolic sector and an elliptic sector ifa2b22 > 0. Taking respectively the Poincare transformations x = 1/z, y = u/z inthe local chart U1 and x = u/z, y = 1/z in the local chart U2, system (ii) aroundthe equator of the Poincare sphere can be written respectively as

u = b21z + (b22 − a2)u2, z = −a2uz(3.2)

and

u = (a2 − b22)u− b21u2z, z = −b22z − b21uz

2.(3.3)

Therefore, there exist singularities of system (ii) located at the infinity of both thex-axis and the y-axis if b22 = a2, which are associated to the origins of system (3.2)and system (3.3) respectively. It is easy to see that the origin of system (3.3) is asaddle if (b22−a2)b22 < 0 and a node if (b22−a2)b22 > 0. Applying [11, Theorem3.5], the origin of system (3.2) is a saddle if (b22−a2)a2 > 0 and its neighborhoodconsists of a hyperbolic sector and an elliptic sector if (b22 − a2)a2 < 0. Whenb22 = a2, the infinity is full up with singularities and there exists a unique orbitconnecting with each point at infinity.

From (2.6) and (2.7), we get that the crossing set LIIc = {(x, y) ∈ R2| y =

0, x = 0} if b21 > 0 and the sliding set LIIs = {(x, y) ∈ R2| y = 0} if b21 < 0

for piecewise smooth system (II). The origin O2 : (0, 0) is the unique singularsliding point of piecewise smooth system (II). Moreover, system (II) has a firstintegral

H+2 (x, y) =

−2b21x+ (a2 − 2b22)y2

x2b22a2 (−2b22 + a2)

(resp. =−b21x lnx+ b22y

2

b22x)

when y ≥ 0 and a2 = 2b22 (resp. a2 = 2b22), and a first integral

H−2 (x, y) =

−2x+ (a2 − 2)y2

x2a2 (−2 + a2)

(resp. =x lnx− y2

x)

when y < 0 and a2 = 2 (resp. a2 = 2). Similar to the research of system (I),the global phase portraits of piecewise smooth system (II) can be obtained bythe global phase portraits of system (ii) in the half planes y > 0 and y < 0together with the dynamics on the crossing set and sliding set, where 64 subcases

10 Y. TANG

correspond to parameter conditions obtained by the signs of b21, b22−a2, b22, a2,a2 and a2 − 1.

In order to research the global dynamics of piecewise smooth system (III), weneed find global dynamics of smooth system (iii). Obviously, the origin O3 : (0, 0)of system (iii) is a saddle if a31b3 < 0 and a node if a31b3 > 0.

In the local charts U1 and U2 of the Poincare sphere, system (iii) becomes

u = (b3 − a31)uz − a32u3, z = −a31z

2 − a32u2z(3.4)

and

u = a32 + (a31 − b3)uz, z = −b3z2,

respectively. Then there exist singularities at infinity of system (iii) only locatedon the x-axis, which are associated to the origin of (3.4), a high degenerateequilibrium. More precisely, the neighborhood of the origin of (3.4) consists oftwo elliptic sectors and one parabolic sector if a31b3 < 0, two hyperbolic sectorsand two parabolic sectors if 0 < a31b3 ≤ 2b23, and two hyperbolic sectors andfour parabolic sectors if a31b3 > 2b23 by applying results of Reyn [31, Figures8.3c-8.3d].

The sliding set of piecewise smooth system (III) is the whole x-axis from (2.8),which is filled with singular sliding points. Except the origin and singularitiesof system (III) located at the infinity of the x-axis, no orbits connect with thepoint in the x-axis from the half planes y > 0 or y < 0. Hence, neither closedorbits nor sliding closed orbits could exist. There exist no homoclinic loops in abounded region, that is, a homoclinic loop has to pass by a singularity at infinityof the x-axis if it exists. Besides, system (III) has a first integral

H+3 (x, y) =

(a31 − 2b3)x+ a32y2

ya31b3 (−2b3 + a31)

(resp. = −a32y2 ln y − b3x

b3y2)

when y ≥ 0 and a31 = 2b3 (resp. a31 = 2b3), and a first integral

H−3 (x, y) =

(a31 − 2)x+ y2

ya31(−2 + a31), (resp. = −y2 ln y − x

y2)

when y < 0 and a31 = 2 (resp. a31 = 2). From an analogous discussion ofsystem (I) in the case b1 < 0, the above analysis of system (III) provides enoughpreparation for studying the global structure of piecewise smooth system (III)and we have its 36 global phase portraits by the signs of a31, b3, a31 − 2b3, a32,a31 and a31 − 2 respectively.

Summarizing the above investigation, we can obtain global dynamics of piece-wise smooth quadratic quasi–homogeneous systems (I)-(III). The proof is com-pleted. �

Here for simplicity, we only present topological phase portraits of piecewisesmooth system (III) (shown in Figure 2) and omit that of systems (I) and (II).The parameter conditions associated to cases (1)-(36) are give in Table 1. Remarkthat we will not consider invertible changes which transform the half plane y > 0into the half plane y < 0 for the topological equivalence of global phase portraits,

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 11

in the sense that the vector fields of piecewise smooth systems are different in thehalf planes y > 0 and y < 0.

Note that for smooth quadratic quasi–homogeneous system (i) (resp. (ii), (iii))there only exists 1 (resp. 4, 3) global phase portrait without taking into accountthe direction of the time, but piecewise smooth quadratic quasi–homogeneoussystem (I) (resp. (II), (III)) has 8 (resp. 64, 36) global phase portraits. Thuspiecewise smooth quadratic quasi–homogeneous systems can exhibit more com-plicated and richer dynamics than the smooth ones.

Figure 2 Parameter conditions

(1) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 < 0 and a31 > 2

(2) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 < 0 and 0 < a31 ≤ 2

(3) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 < 0 and a31 < 0

(4) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 > 0 and a31 > 2

(5) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 > 0 and 0 < a31 ≤ 2

(6) a31 < 0, b3 < 0, a31 ≥ 2b3, a32 > 0 and a31 < 0

(7) a31 < 0, b3 < 0, a31 < 2b3, a32 > 0 and a31 > 2

(8) a31 < 0, b3 < 0, a31 < 2b3, a32 > 0 and 0 < a31 ≤ 2

(9) a31 < 0, b3 < 0, a31 < 2b3, a32 > 0 and a31 < 0

(10) a31 < 0, b3 < 0, a31 < 2b3, a32 < 0 and a31 > 2

(11) a31 < 0, b3 < 0, a31 < 2b3, a32 < 0 and 0 < a31 ≤ 2

(12) a31 < 0, b3 < 0, a31 < 2b3, a32 < 0 and a31 < 0

(13) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 < 0 and a31 > 2

(14) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 < 0 and 0 < a31 ≤ 2

(15) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 < 0 and a31 < 0

(16) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 > 0 and a31 > 2

(17) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 > 0 and 0 < a31 ≤ 2

(18) a31 > 0, b3 > 0, a31 ≤ 2b3, a32 > 0 and a31 < 0

(19) a31 > 0, b3 > 0, a31 > 2b3, a32 < 0 and a31 > 2

(20) a31 > 0, b3 > 0, a31 > 2b3, a32 < 0 and 0 < a31 ≤ 2

(21) a31 > 0, b3 > 0, a31 > 2b3, a32 < 0 and a31 < 0

(22) a31 > 0, b3 > 0, a31 > 2b3, a32 > 0 and a31 > 2

(23) a31 > 0, b3 > 0, a31 > 2b3, a32 > 0 and 0 < a31 ≤ 2

(24) a31 > 0, b3 > 0, a31 > 2b3, a32 > 0 and a31 < 0

(25) a31 > 0, b3 < 0, a32 < 0 and a31 > 2

(26) a31 > 0, b3 < 0, a32 < 0 and 0 < a31 ≤ 2

(27) a31 > 0, b3 < 0, a32 < 0 and a31 < 0

(28) a31 > 0, b3 < 0, a32 > 0 and a31 > 2

(29) a31 > 0, b3 < 0, a32 > 0 and 0 < a31 ≤ 2

(30) a31 > 0, b3 < 0, a32 > 0 and a31 < 0

(31) a31 < 0, b3 > 0, a32 < 0 and a31 > 2

(32) a31 < 0, b3 > 0, a32 < 0 and 0 < a31 ≤ 2

(33) a31 < 0, b3 > 0, a32 < 0 and a31 < 0

(34) a31 < 0, b3 > 0, a32 > 0 and a31 > 2

(35) a31 < 0, b3 > 0, a32 > 0 and 0 < a31 ≤ 2

(36) a31 < 0, b3 > 0, a32 > 0 and a31 < 0

Table 1. Parameter conditions of Figure 2.

12 Y. TANG

x xx

y y y

OOO

( )1( )1 ( )2 ( )3

x

y y y

( )1( )4 ( )5 ( )6

xxx

O OO

x

y y y

( )7( ( )8 ( )9

xxx

OO

O

x

y y y

( )10 ( )11 ( )12

xxx

OO

O

x

y y y

( )13 ( )14 ( )15

xx x

OOO

x

y y y

( )16 ( )17 ( )18

x

OOO

x

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 13

y y y

( )19 ( )20 ( )21

x

OOO

x x

y y y

( )22 ( )23 ( )24

x

OOO

x x

y y y

( )25 ( )26 ( )27

xOO

O

x x

y y y

( )28 ( )29 ( )30

xOOO x x

y y y

( )31 ( )32 ( )33

xOOOx x

y y y

( )34 ( )35 ( )36

xOOOx x

Figure 2. The global phase portraits of system (III).

14 Y. TANG

From the global dynamics of piecewise smooth systems (I)-(III), we can re-search the global bifurcation and unfolding of all special orbits including ho-moclinic loops (or heteroclinic loops), closed orbits, equilibria and equilibria atinfinity for the systems. As seen in Figure 2 for example, the unfoldings arepresented in phase portraits (2) and (3) if we choose µ1 = a31 as an unfoldingparameter. A homoclinic (or heteroclinic) bifurcation happens when µ1 passesthrough zero. More precisely, when µ1 > 0 there exist infinitely many heteroclinicloops connecting with the origin and equilibria at infinity on the x-axis, and whenµ1 < 0 some heteroclinic orbits in the half plane y < 0 become homoclinic loopsconnecting with the equilibria at infinity on the x-axis. If we choose µ2 = a32 asanother unfolding parameter, we find that the heteroclinic loops burst out whenµ2 varies from negative to positive by phase portraits (2) and (5) in Figure 2. Atthe same time it can be observed clearly the change of sectors in a neighborhoodof equilibria and equilibria at infinity, which exhibits bifurcations of equilibria.We can also notice other global and local bifurcations if we choose b3, a31 − 2b3,a31 and a31 − 2 as unfolding parameters for piecewise smooth system (III).

4. Limit cycle bifurcations by perturbing piecewise smoothquadratic quasi–homogeneous systems

From Theorem 2, only system (I) of all piecewise smooth quadratic quasi-homogeneous systems has a center at the origin, which is global if it exists. In thissection we research the bifurcation of limit cycles by perturbing piecewise smoothquadratic quasi–homogeneous system (I) with arbitrary piecewise polynomials ofdegree n ∈ N, where N = Z+ ∪ {0}.

Consider the following one-parametric family of piecewise smooth systems

x = Hy(x, y) + ϵf(x, y), y = −Hx(x, y) + ϵg(x, y),(4.1)

where ϵ ∈ R is the small perturbation parameter,

H(x, y) =

{H+(x, y) := H+

1 (x, y) = a13 y

3 − b12 x

2, if y ≥ 0,

H−(x, y) := H−1 (x, y) = a1

3 y3 − 1

2x2, if y < 0,

f(x, y) =

{f+(x, y) =

∑ni+j=0 c

+ij xiyj , if y ≥ 0,

f−(x, y) =∑n

i+j=0 c−ij xiyj , if y < 0,

g(x, y) =

{g+(x, y) =

∑ni+j=0 d

+ij xiyj , if y ≥ 0,

g−(x, y) =∑n

i+j=0 d−ij xiyj , if y < 0

for arbitrary c±ij , d±ij ∈ R, and a1 < 0, b1 > 0 and a1 > 0. Our aim is to give

the maximum number of limit cycles in terms of n which can bifurcate from theperiodic orbits of the center at the origin of system (I) with ϵ = 0, inside thefamily (4.1) for nonzero ϵ.

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 15

We will use Melnikov method to investigate the number of bifurcated limitcycles from system (4.1). Let

L+h = {(x, y) ∈ R2 | H+

1 (x, y) = −h

2, h > 0}, if y ≥ 0,

L−h = {(x, y) ∈ R2 | H−

1 (x, y) = − h

2b1, h > 0}, if y < 0.

Then the family of periodic orbits of system (4.1) with ϵ = 0 is presented byLh = L+

h ∪ L−h , where h > 0.

Using the idea in [27] for piecewise smooth system with a discontinuous linethe y-axis, we have the first order Melnikov function for system (4.1) along thefamily of periodic orbits Lh, which is

M(h, ϵ) =H+

x (A)

H−x (A)

(H−x (B)

H+x (B)

∫L+h

(g+dx− f+dy) +

∫L−h

(g−dx− f−dy)),(4.2)

where points A = (√

hb1, 0) and B = (−

√hb1, 0), as shown in Figure 3.

OB

A

Figure 3. The closed orbit of system (I) and its perturbation.

Lemma 4. For piecewise smooth system (4.1), we have the first order Melnikovfunction

M(h, ϵ) =

n∑2k+j=0

ξ2k,j hk+j3+ 1

2 ,(4.3)

where coefficients ξ2k,j are given in (4.9).

Proof. Firstly, we compute that H+x (A) = −b1

√hb1, H−

x (A) = −√

hb1, H−

x (B) =√hb1

and H+x (B) = b1

√hb1

in (4.2).

16 Y. TANG

Restricted on L+h and L−

h , we solve y = 3

√3

2a1(−h+ b1x2) and y = 3

√3

2a1(− h

b1+ x2),

respectively. Then for i, j ∈ N we calculate

∫L+h

xiyjdx = ( 32a1

)j3

∫ −√

hb1√

hb1

xi(−h+ b1x2)

j3dx

= −2( 32a1

)j3

∫√hb1

0 x2k(−h+ b1x2)

j3dx

= d+2k,j hk+j3+ 1

2 ,

(4.4)

∫L+h

xiyjdy = b1a1

∫L+h

xi+1yj−2dx

= b1a1( 32a1

)j−23

∫ −√

hb1√

hb1

xi+1(−h+ b1x2)

j−23 dx

= −2 b1a1( 32a1

)j−23

∫√hb1

0 x2k+2(−h+ b1x2)

j−23 dx

= c+2k+1,j hk+j+13

+ 12 ,

(4.5)

∫L−h

xiyjdx = ( 32a1

)j3

∫√hb1

−√

hb1

xi(x2 − hb1)j3dx

= 2( 32a1

)j3

∫√hb1

0 x2k(x2 − hb1)j3dx

= d−2k,j hk+j3+ 1

2

(4.6)

and ∫L−h

xiyjdy = 1a1

∫L−h

xi+1yj−2dx

= 1a1( 32a1

)j−23

∫√hb1

−√

hb1

xi+1(x2 − hb1)j−23 dx

= 2 1a1( 32a1

)j−23

∫√hb1

0 x2k+2(x2 − hb1)j−23 dx

= c−2k+1,j hk+j+13

+ 12

(4.7)

where k ∈ N,

d+2k,j = −2( 32a1

)j3 ( 1

b1)k+

12

∫ 10 x2k(−1 + x2)

j3dx,

c+2k+1,j = −2 b1a1( 32a1

)j−23 ( 1

b1)k+1+ 1

2

∫ 10 x2k+2(−1 + x2)

j−23 dx,

d−2k,j = 2( 32a1

)j3 ( 1

b1)k+

j3+ 1

2

∫ 10 x2k(−1 + x2)

j3dx,

c−2k+1,j = 2 1a1( 32a1

)j−23 ( 1

b1)k+

j+13

+ 12

∫ 10 x2k+2(−1 + x2)

j−23 dx.

(4.8)

Notice that d+2k,j c+2k+1,j d

−2k,j c

−2k+1,j = 0,

∫L±h

xiyjdx = 0 for odd i and∫L±h

xiyjdy =

0 for even i.Substituting (4.4)–(4.7) into the formula (4.2), we obtain the Melnikov function

of system (4.1) as

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 17

M(h, ϵ) = b1

( 1

b1

∫L+h

(g+dx− f+dy) +

∫L−h

(g−dx− f−dy)),

=

∫L+h

( n∑i+j=0

d+ij xiyjdx−n∑

i+j=0

c+ij xiyjdy)

+ b1

∫L−h

( n∑i+j=0

d−ij xiyjdx−n∑

i+j=0

c−ij xiyjdy)

= h12

( n∑2k+j=0

d+2k,j d+2k,j hk+j3 −

n∑2k+1+j=0

c+2k+1,j c+2k+1,j hk+j+13

)+ b1h

12

( n∑2k+j=0

d−2k,j d−2k,j hk+j3 −

n∑2k+1+j=0

c−2k+1,j c−2k+1,j hk+j+13

)= h

12

( n∑2k+j=0

d+2k,j d+2k,j hk+j3 −

n∑2k+j=0

c+2k+1,j−1 c+2k+1,j−1 hk+j3

)+ b1h

12

( n∑2k+j=0

d−2k,j d−2k,j hk+j3 −

n∑2k+j=0

c−2k+1,j−1 c−2k+1,j−1 hk+j3

)= h

12

n∑2k+j=0

ξ2k,j hk+j3 ,

where k ∈ N,

ξ2k,j = d+2k,j d+2k,j − c+2k+1,j−1 c+2k+1,j−1

+b1d−2k,j d−2k,j − b1c

−2k+1,j−1 c−2k+1,j−1

(4.9)

and d+2k,j , c+2k+1,j , d

−2k,j , c

−2k+1,j are displayed in (4.8). Therefore, (4.3) is proved.

�

In order to determine how many limit cycles the piecewise smooth system (4.1)can have, we analyze zeros of Melnikov function (4.3). For convenience, we set

h = h6 and get from (4.3) that

(4.10) M(h, ϵ) = h3n∑

2k+j=0

ξ2k,j h6k+2j = h3n∑

i+j=0

ξi,j h3i+2j ,

where i is even.The zero problem of M(h, ϵ) is transferred to determine the cardinal of the set

S(n) = {3i+ 2j : 0 ≤ i+ j ≤ n, i even, i, j ∈ N}.That is, we need to find how many different elements exist in the set S(n).

We denote a trapezoid by

Υ1(n) = {(i, j) : 0 ≤ i+ j ≤ n, j < 3, i even, i, j ∈ N},

18 Y. TANG

and a set by

Υ(n) = {3i+ 2j : 0 ≤ i+ j ≤ n, j < 3, i even, i, j ∈ N}.The cardinal of the set S(n) is given in the following lemma by applying a similarideal in [18] for smooth quasi–homogeneous polynomial differential systems.

Lemma 5. S(n) = Υ(n).

Proof. Obviously we have S(n) ⊃ Υ(n). We only need to prove that S(n) ⊂ Υ(n).Let 3i+2j be an arbitrary element in S(n) such that j ≥ 3 and i is even. Then

there exists ι ∈ Z+ satisfying 3ι ≤ j < 3(ι+ 1). We have

3i+ 2j = 3(i+ 2ι) + 2(j − 3ι) = 3i1 + 2j1,

where even i1 = i+ 2ι ≥ 0, 0 ≤ j1 = j − 3ι < 3 and 0 ≤ i1 + j1 = i+ j − ι ≤ n,implying 3i1 + 2j1 ∈ Υ(n) and 3i+ 2j ∈ Υ(n). Hence S(n) ⊂ Υ(n). �

Remark that from Lemma 5 all the values of 3i + 2j for the degrees of h in(4.10) are taken exactly by points on the trapezoid Υ1(n).

We need use the following version of the Descartes Theorem proved in [6] tojudge real zeros of the Melnikov function.

Theorem 6 (Descartes theorem). Consider the real polynomial q(x) = ai1xi1 +

ai2xi2 + . . . + airx

ir with 0 = i1 < i2 < . . . < ir. If aijaij+1 < 0, we say thatwe have a variation of sign. If the number of variations of signs is m, then thepolynomial q(x) has at most m positive real roots. Furthermore, always we canchoose the coefficients of the polynomial q(x) in such a way that q(x) has exactlyr − 1 positive real roots.

Let Ξ(n) denote the maximal number of limit cycles, which are produced inpiecewise smooth system (4.1) and bifurcated from the period solutions of piece-wise smooth quadratic quasi–homogeneous system (I) by taking into account thezeros of the first order Melnikov function.

Theorem 7. For piecewise smooth quadratic quasi–homogeneous system (I) per-turbed inside the class of all piecewise smooth polynomial differential systems ofdegree n when a1 < 0, b1 > 0 and a1 > 0, the number Ξ(n) = 2[n+1

2 ] + [n−12 ] − 1

(resp. Ξ(n) = 2[n2 ]+[n+22 ]−1) if n is odd (resp. even) by the first order Melnikov

function. Moreover, there exist perturbations of piecewise smooth polynomial sys-tems of degree n in (4.1) with exactly Ξ(n) limit cycles.

Proof. From the expression of the first order Melnikov function (4.10) and Theo-rem 6, we get that Ξ(n) is equal to |S(n)| − 1, where |S(n)| is the cardinal of theset S(n). Applying Lemma 5 we have Ξ(n) = |Υ(n)| − 1.

Besides, all the values of the function 3i+ 2j are different for different pointson the trapezoid Υ1(n). In fact, if 3i + 2j = 3i + 2j for both (i, j) and (i, j) inΥ1(n), we have 3(i− i) = 2(j − j), yielding that 3|(j − j). Because 0 ≤ j, j < 3,we get j − j = 0 and furthermore i = i. Therefore, each of all values of the setΥ(n) is taken exactly once by one point on the trapezoid Υ1(n).

DYNAMICS AND UNFOLDING OF PIECEWISE QUASI–HOMOGENEOUS SYSTEM 19

Taking j = 0, 1, 2 respectively, we calculate

|Υ(n)| = [n+ 1

2] + [

n+ 1

2] + [

n− 1

2] = 2[

n+ 1

2] + [

n− 1

2]

if n is odd and

|Υ(n)| = [n+ 2

2] + [

n

2] + [

n

2] = 2[

n

2] + [

n+ 2

2]

if n is even. Thus, the formula of Ξ(n) in this theorem is proved.In addition, we notice that the Melnikov function (4.10) has Λ := |Υ(n)| terms

with different degrees of h, whose coefficients are denoted by

ξ1, ξ2, ..., ξΛ.

These coefficients are linear combinations of parameters d+2k,j , c+2k+1,j−1, d

−2k,j and

c−2k+1,j−1, as seen in (4.9). It reveals that the matrix

∂(ξ1, ξ2, ..., ξΛ)

∂(c+1,0, c+3,0, ..., c

+2k+1,j−1, ..., c

−2k+1,j−1, ..., d

+2k,j , ..., d

−2k,j , ...)

has a full row rank. So there exists an array

(c+1,0, c+3,0, ..., c

+2k+1,j−1, ..., c

−2k+1,j−1, ..., d

+2k,j , ..., d

−2k,j , ...)

satisfying that the Melnikov function M(h, ϵ) in (4.10) has Λ − 1 variations of

signs. By Theorem 6, the function M(h, ϵ) has exactly Ξ(n) positive zeros. Thenwe obtain that the Melnikov function in (4.3) has exactly Ξ(n) positive zeros andpiecewise smooth system (4.1) has Ξ(n) limit cycles bifurcating from the periodicsolutions of the piecewise smooth quadratic quasi–homogeneous center of system(I) by using the first order Melnikov function. �

Acknowledgements

The author has received funding from the European Union’s Horizon 2020research and innovation programme under the Marie Sklodowska-Curie grant a-greement No 655212, and is partially supported by the National Natural ScienceFoundation of China (No. 11431008). The author thanks Professor Valery Ro-manovski for fruitful discussions on the work.

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1 Center for Applied Mathematics and Theoretical Physics, University of Mari-bor, Krekova 2, Maribor, SI-2000 Maribor, Slovenia

2 School of Mathematical Science, Shanghai Jiao Tong University, DongchuanRoad 800, Shanghai, 200240, P.R. China

E-mail address: mathtyl@sjtu.edu.cn